But now we've got something very significant because now we can actually see the principle that's called the Law of Gravity almost staring us in the face. Here's the idea, go back to the center of gravity principle. One thing I already told you, you've got to have the ratios of the distances from the center of gravity remain constant for them to be in balance.

What's the other thing you do? Well this is straight out of Archimedes. The ratio of the two distances is the ratio of the two weights when they're hanging down. But now we're looking at weight of Jupiter and the weight of the sun. What are we talking about with weight?

Well, presumably, we're talking about what they would weigh at the surface of the earth. What would Jupiter weigh terrestrially falling toward the Earth? What would the Sun weigh falling toward the Earth? But we already know something. Weight is going to change depending on distance. Weight is not a universal quantity.

It's variable depending on inverse square how far away you are with gravity. So we would like a quantity to replace weight, we'll get to that in a moment, a quantity to replace weight that's invariant. Before we go to that though, we already had this earlier, what's the ratio of the two distances?

It's to the two forces. Now we can say something else from this relation. The two forces have to be proportional to their respective weights. So the attractive force towards Jupiter, this is concluding from the center of gravity principle, is proportional to its weight at the surface of the Earth.

The attractive force towards The Sun, Helios, is proportional to its weight to toward the center of the Earth. And from that, if you let me substitute m for quantity of matter up here, I can get two ways to derive the law of gravity. This was the discovery I made in 1991 in this course that led to the paper, How Newton Discovered Universal Gravity, then in the remaining next half hour I'm gonna be modifying because I know more now than I knew in 1991.

But one way to do it is simply now drop the weights in that relationship and have it be the masses. So that means that the two invariant quantities that are the majors of the centripetal tendencies towards Jupiter and the Sun are proportional to the quantities of matter in Jupiter and the Sun.

That's what that new definition of pondus gives us. I could have put pondus up there. And we know already what the effect of any force is going to be, namely it's going to be the tendency over the inverse square of the distance. So therefore the effect on Jupiter of the force toward the Sun is going to be the mass of the Sun divided by the square of the distance between them.

But we also know that a force acts, the force is proportional to the change of the motion and the change of the motion has to include Jupiter's mass as well. Okay? So this just gives us what the total force is, what Newton came to be called this, the motive force, on Jupiter.

And by like reasoning, we get the motive force on the Sun towards Jupiter. And notice we have the masses of each of them in there, and they're equal to one another. So the third law of motion turns out to hold for this. And thereby, we get the law of gravity.

A second way to get it is to simply say, what are the forces? The forces are a cubed over p squared over the inverse square relation. The product of that times the mass of Jupiter is the force acting on Jupiter. Similarly the product of those two, the mass of the Sun and the force towards Jupiter is the force of the Sun, I have to do this carefully, the force of Jupiter on the Sun.

Assume the third law of motion holds, so those two forces have to be equal. And, once again, you arrive right away at the fact that the tendencies toward the central bodies are proportional to the masses in the central bodies. And you get the law of gravity once again.

So this is a way Newton could have arrived at the law of gravity where the peculiar part of it, that I haven't shown as right so far, is why should the tendency toward the sun be proportional to the quantity of matter in the sun. So far all I've said is the force acting on the body is proportional to the quantity of matter in it.

What we're here getting is the symmetry relation. And I'm popping it out this way, applying the third law of motion or going the other direction once I've made the distinction between weight and mass. Now, this is not universal gravity. This is the force between Jupiter and the Sun.

I'm gonna amplify on this significantly. When I say this view is now the accepted view of how Newton got to universal gravity, what I mean is in a later article Kurtis Wilson said this the right story, and nobody's been disputing him. I'm about to dispute myself by showing you that this story is more complicated than that in a very interesting way, but I hadn't looked at the thing I'm about to go to now.

And this is the first time most people will have ever seen this. I've been working for 18 months whenever I have free time, and Sammy has been assisting me, on this, on a document that's called De Motu Corporum Liber Secundus. It is 55, 56 pages long. Newton describes it later as written in the popular style.

Now you've seen Descartes, except you saw Descartes, well I showed it to you in Latin. How was Descartes' Principia written? In numbered articles with apostle in the margin, summarizing what's in the article. How is this written? This is in Humphrey Newton's hand. That's in Issac's hand. Issac numbers them and puts a postal, telling you what each article is doing.

The popular style means the very style in which Descartes wrote his Principia. So this was written to mimic Descartes' Principia. I now feel, well a comment, a version of this got published posthumously after Newton died, called The System of the World. It was published both in Latin and in English.

And as a result, almost nobody ever went back and looked at the original because they assumed what was published in 1728 was accurate. In fact, what was published in 1728 has two drawbacks. The first drawback is all modifications made in the original are ignored. And of course those are very instructive cuz they show Newton change.

Second, and more important they decided to make it look more like the Principia. So they changed proposition numbers and various claims. To make it appear consistent with the Principia. When they did that, they totally misled when this was written. In particular, they referred the propositions that didn't exist yet at the time the original was written.

And you go back and look at the original number and you can tell that. So what has happened here is when Ann Whitman and Bernard Cohen were putting together what's called the Variorum Edition of the Principia, which has all the variations and all the different editions. Ann did a transcript of this and an initial translation, including all the deletions, etc., and it sat there from the late 1970's until late 1990s when Bernard said, we out to take this and do something with it, but then he died, and I just had other things to do.

So for, it's a very complicated reason, somebody asked me what happened in 1685. You'll find out about this later. And I finally started looking at this and realized this is a gold mine of a document. It tells us more about how Newton got to the Principia than everything else put together, because it's intact and has all the deletions.

Okay? And it's long. It's the same lentgh as Book Three of the Principia and it covers the same material at an earlier stage. Okay, so what I'm gonna be showing you is from that. There's a lot of work still to be done on this. I'm hopeful that sometime by next summer, I'll have the critical edition of this done with both the full text, actually, the full text in Latin and all of the variations are done.

Other than minor corrections by Samian and what remains to be done is to put out a really clean English translation with a lot of annotations of it. But, anyway, I'm going to be showing you portions of this. What this does is it starts out by establishing the inverse square.

In an argument that has some resemblance to the one you'll see in the Principia, I'm not worried about that now. He gets to the inverse square. Then he says, it looks like the magnitude of the forces, he's doing it off of the three hash power rule the way I've been doing it all this time here.

The magnitude of the forces towards Jupiter, Sun etc., are proportional to their sizes. And he has a section on what the precise sizes of them are. At that point, we pick up, that's where I'm picking up, with a pair of articles, 18, 19. And since you've never seen these, I'm going to read them.

The red is only because I'm, for emphasis. A second agreement between the forces and the attracted bodies is akin to the one just described. Notice at the top. Another agreement between forces and bodies, it is proved for heavenly bodies. What's the prior agreement? Between the forces and the sizes of the bodies.

So he's gonna view a different one now. Since the action of the centripetal force upon the planets decreases in the duplicate ratio of the distance and the periodic time is increased in sesquialtera ratio, it is manifest that if equal planets were equally distant from the Sun, their actions would be equal and their periodic times would be equal.

And then if unequal, planets at equal distances, their collective actions would be as the bodies of the planets. Collective action, I don't know exactly what he means. The word was inserted, as you can see, to clarify what their actions would be. So he's saying they would be proportional to the bodies of the Sun.

And notice what he's done here, he originally said would be as the pondera of the planets and he comes back and looks at this article as it was written and he realizes something. Ponderem here has to mean weight. Everywhere else it's referring to quantity of matter. And he decides, uh-oh, pondus is not the right word to use for this.

So at this juncture he scratches out every occurrence of pondus and he does it through the rest of this document where it means what I was calling heaviness before, and replaces it by expressions like quantity of matter. He hasn't got mass yet, but he's decided the word he originally wanted for it was lousy, okay.

Continuing from there. For actions that were not as the bodies, to be moved could not draw those bodies equally back from the tangents of the orbits and caused revolutions to completed in equal times in orbits that are also equal. Now what am I trying to stress here? It's essentially a point I made earlier.

If you look at the proof of the three hash power rule for elliptical orbits, nothing is said about the mass of the moving body. He's essentially assuming all bodies experience the same centripetal acceleration toward the Sun at the same point, and that's what he's saying here. He actually says that.

Therefore the force of the Sun has to be proportional to the matter in the bodies. That's not the key one, though. I'm now gonna turn to the key one. But neither could the motions of the satellites of Jupiter be so regular If the circumsolar force were not exerted equally upon Jupiter and all the satellites in proportion to their weights.

That's what he needs weights for. Now pause for a moment. What did Flamsteed ask him? Why is the motion of the satellites of Jupiter so regular? Newton knows the Sun is acting on all of them. Right? That's what keeps them all in orbit together. Surely, the quantity of matter in each of the satellites is different from the quantity of matter in Jupiter.

But the Sun's effect is the same on all of them. How can that be, unless the sun, as a force, is adjusting itself to each of those bodies, so that they all fall equally towards the Sun? But that can only be so if the force of the Sun on each of the bodies is proportional to the quantity of matter in the bodies.

I'm showing you what caused him to do the experiment. It's a celestial wary that bothered him. It wasn't Galileo saying all bodies fall at the same rate. It was his realization that that has to be true for Jupiter and his satellites. They have to be falling at the same rate.

Okay? So, we'll just continue now. And the same is true of Saturn and its satellite, and also of the Earth and our Moon. We'll worry next semester about proposition 35, corollary two and three. And is manifest, and soon will be made for, scratched out, at equal distances therefore, there is an equal action of centripetal force.

Upon all the planets in proportion to their bodies or quantities of matter in the bodies, and that's also upon all the particles of that quantity of which the planets are composed. Or otherwise, they would start coming apart, because the force would act differentially. For if the action were greater upon particles of one kind of matter and less upon those of another, then in proportion to the quantity of matter, the action upon the planets would also be greater or less.

Not only in proportion to the quantity, but also in accordance with the kind of matter which would be found more abundantly in one body and more sparingly in another. So this is a comment about the actual material doesn't matter. Remember Descartes said it does matter. The reason one kind of material is more dense than another is one has more solid matter in it.

He's saying forget that, this is the same for all bodies. Now we get, it is proof for terrestrial bodies. So he's just done it for celestial bodies, now he's gonna do it for terrestrial bodies. I have actually tested this proportion with the greatest exactness as possible in different kinds of bodies that exist on our Earth.

So you now see why he did the experiment. The action of a circumterrestrial force that is proportional to the bodies to be moved, will move them in equal times with equal velocity by law two. And will make all bodies that are let fall descend through equal spaces and equal times.

And will also make all bodies suspended by equal cords oscillated equal times. If the action is greater the times will be smaller etc. Others have long since observed that all bodies descend in equal times at least if the very small resistance of air is removed. And it is possible to discern the equality of the times to the highest degree of accuracy in pendulums.

I have tested this with gold, silver, the same sequence. I got two equal wooden boxes. I filled one with with wood, and I suspended the same weight as exactly as I could at the center of oscillation of the other. The boxes hanging by equal 11 foot chord, made pendulums exactly like one another with respect to their weight, shape and air resistance.

Okay? The bodies have the same shape. Therefore any motion of air, if they're moving in unison with one another, the air resistance will be the same. So all the forces acting on it are going to be the same if they're moving together. Then when placed close to each other, they kept swinging back and forth together with equal oscillations for a very long time.

Accordingly the amount of matter in the gold, by proposition, he hasn't got a number yet, was to the amount of matter in the wood as the action of the mode of force. That's a key term, mode of force, the force producing the motion upon all the gold to this action upon all the wood, that is, is the weight of one to the weight of the other, and so for all the others.

In these experiments and bodies are the same weight, a difference of matter that would have been even less than a thousandth of a whole could have been clearly noticed. Now deleted, because of this agreement I have throughout designated the quantity of matter in each body, by the word pondus, using the name of the measure of the thing measured, as his common.

And of course he's now deleted that. Cuz he's not using pondus. He's gotta come up with a new word. One part in a thousand. When Curtis Wilson redid this experiment, he concluded it's good to within one part of 2,000, Newton was being conservative. By the time Einstein started, by the time he got near working on this in 1905, Otvos, a Hungarian experimentalist had confirmed five parts in 10 million, excuse me 100 million.

Five parts in 100 million. This whole's to within five parts in 100 million. And it is the backbone of the Theory of General Relativity. I'm gonna argue it's the backbone of Newton's theory too, because this is a mysterious claim. Go back to the principle of inertia. What's it say?

It says an external force is acting as a cause. What sort of external cause adjusts itself to the body on which it's acting? That's strange. That's even weird. If it's an external force, the magnitude of that force should just be there. It shouldn't, in any way adjust in magnitude to which body it's acting on.

That's what it is to call it external, external to the body. It's independent of the body. He's saying to the contrary here, gravity doesn't work that way, nor does the centripetal force celestially work that way. They both violate this principle. Conclusions he draws from it. Initially, the unanimity of the agreements, and since the action of centripetal force upon the attracted body at equal distances, is proportional to the matter of this body, it is reasonable also to grant.

I want you to hear that phrase. That's not saying I have a proof of. It is reasonable also to grant that it is proportional as well to the matter in the attracting body. Notice the word here attracting, it's being translated, two different words are constantly translated all through Newton's stuff as attracting.

Traha, trahanta in this case means to draw something. So if I were to pull a boat I would use the trahanta. If i were having something that's attracted to me I would use atrahanta. Here initially he's saying the attracting body, the body that's drawing you toward it. For the action is mutual and causes the bodies, by a mutual endeavor by law three, to approach each other and accordingly the action in one body must necessarily be in conformity with the action in the other.

One body can be considered as attracting, the other is attracted, but this distinction is more mathematical than natural. The attraction is really that of either of the two bodies towards the other, and thus is the same kind in each of the bodies. So that's his first way of getting to the law of gravity that I described before.

Now what follows and their coincidence, the coincidence of the two agreements. I'm not sure what he means by their coincidence, that's the word he used. The margin note is et cointed condintia. Now this is the remarkable paragraph that I'm leading to and nothing like this appears in the Principia, nothing remotely like it.

And hence it is that the attractive force is found in both bodies. The sun attracts Jupiter and the other planets, Jupiter attracts its satellites and similarly the satellites act on one another and on Jupiter, and all the planets act on one another. And although in a pair of planets, the action of each on the other can be distinguished and can be considered as paired actions by which each attracts the other.

Yet in as much as these are actions between two bodies, they are not two but a simple operation between two termini. Two bodies can be drawn to each other by the contraction of a single rope between them. The cause of the action is two fold, namely the disposition of the two bodies.

The action is likewise two fold in so far as it is upon two bodies, but in so far as it is between two bodies, it is a simple and single action. There is not for example, one operation by which the Sun attracts Jupiter and another operation by which Jupiter attracts the Sun, but a single operation by which the Sun and Jupiter endeavor to approach each other.

By the action by which the Sun attracts Jupiter, Jupiter and the Sun endeavor to approach each other. By law three and by the action which Jupiter attracts the sun. Jupiter and the Sun also endeavor to approach each other. Moreover, the sun is not attracted to a two-fold action towards Jupiter.

Nor is Jupiter attracted by a two fold action toward the sun, but there is one action between them by which they both approach each other. Iron attracts a lodestone just as much as a lodestone attracts iron. For, any iron in the vicinity of a loadstone attracts other iron, also.

But, the action between the loadstone and the iron is simple. And natural philosophers consider it as simple. The operation of the iron upon the lodestone is the very operation of the lodestone between itself and the iron, by which both endeavor to approach each other. This is manifest from the fact that, if the lode stone is removed, nearly the whole force of the iron ceases.

In this way, conceive that a simple operation arising from the concurring nature of two planets is exerted between them. Then, this operation will be the same with respect to both, and thus, being proportional to the matter in one of them, will be proportional to the matter in the other.

Okay, so, it's not an external cost. It's not an external cost adjusting itself to different bodies. It's one thing between two bodies. Okay? Now that's a wild proposal at this time. That's one reason nothing like this appears anywhere in the Principia. The other thing to notice is how redundant this is.

He knows how wild the proposal is. He's trying to make it intelligible. But he's also recognizing, that's what the law of gravity is screaming at him. Both terms, both masses are in there because it's not the action of one body on another. Don't think of it that way at all.

Think of it as a single interaction between two bodies. And now he's about to conclude, not just between two bodies, between every two bodies everywhere, in fact between every two particles of matter. Now, I don't want to keep you and I want to finish material at the end of the course, so let me just jump.

What follows, and I have elliptcized it Is an explanation for why we can't experience this. I mean, he's just said the Earth is attracted toward me. So if I jump, the Earth is going to be attracted toward me as I fall back toward the Earth. We can't experience this, we can't experience any of the actions of our bodies toward one another.

So what he says, someone will perhaps say that by this law all bodies must attract each other, which is contrary to experience in terrestrial bodies. But my answer is that there is no experience at all in terrestrial bodies. Not even whole mountains would suffice for sensible effects. At the foot of a hemispherical mountain three miles high, and six miles wide, a pendulum attracted by the force of a mountain will not deviate two minutes from the perpendicular.

It is possible to observe these forces only in the huge bodies of the planets, but we can discuss lesser bodies as follows. And what does follow is a long discourse that starts with a thought experiment about if the Earth were divided into three pieces, let's do two pieces, and they were not equal.

If they did not attract to each other equally In accord with the third law of motion. The bigger one would make the smaller one, the net effect is they would create motion. They have to be in balance with one another. It's a thought experiment. That thought experiment shows up in the Principia, first time in the second edition.

You'll be reading it for the reading for the first class, but it's not there in the first edition. Then, in the middle of that, Newton inserts a claim, I'm not going to read it out loud to save time, that it can't be the gravities aim toward an empty place in space.

So if you move the sun away. Jupiter won’t continue going toward that spot where the sun was before, okay? And then it continues, I dont show the whole thing, the entire thing is cancelled. Here's the part in Newton's hand that's cancelled. It's written in such a way that it appears have been dictated.

With the insert put in and then on the separate sheet and then Newton cancelled. Nothing like this appears in the Principia, he never tries to argue that all terrestrial bodies are attracted to one another. He basically just asserts it, drawing the conclusion from the third law of motion and the law of gravity.

But when you see him working it out here, this is how he gets to universal gravity. He gets there from the idea that if gravitational forces adjust themselves to the mass of the bodies, that makes no sense unless it's a mutual action. In which the masses enter reciprocally with respect to one another.